Free Resolutions

这个项目涉及交换代数的研究。本课程的核心目标是理解多项式方程系统的解，可能包含大量变量和大量方程。与此密切相关的是自由分辨率的概念。构建这样的解决方案相当于反复求解多项式方程组。多年来，最小自由分辨率一直是交换代数的中心对象和富有成效的工具。希尔伯特在1890年的一篇著名论文中提出了构建自由分辨率的思想。对这些物体的研究在20世纪下半叶蓬勃发展，最近取得了惊人的进展。这个领域非常广泛，与其他数学领域有很强的联系和应用。该项目的更广泛影响包括撰写说明性论文，为本科生组织专业发展研讨会，以及组织数学会议。本课题的主要研究目标是在理解最小自由分辨率的结构及其数值不变量方面取得重大进展。特别地，PI将：与M. Mastroeni和J. McCullough共同研究科祖尔代数；继续研究二项边缘理想的最小自由分辨率；研究外代数上最小自由分辨率的渐近结构。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project concerns research in Commutative Algebra. A core goal in the subject deals with understanding the solutions of a system of polynomial equations, possibly in a large number of variables and with a large number of equations. Closely related to this is the concept of a free resolution. Constructing such a resolution amounts to repeatedly solving systems of polynomial equations. For many years, minimal free resolutions have been both central objects and fruitful tools in Commutative Algebra. The idea of constructing free resolutions was introduced by Hilbert in a famous paper in 1890. The study of these objects flourished in the second half of the twentieth century and has seen spectacular progress recently. The field is very broad, with strong connections and applications to other mathematical areas. The broader impacts of the project include the writing of an expository paper, organization of professional development workshops for undergraduate students, and organization of mathematical conferences.The main research goal in this project is to make significant progress in understanding the structure of minimal free resolutions and their numerical invariants. In particular, the PI will: work jointly with M. Mastroeni and J. McCullough on Koszul Algebras; continue work on minimal free resolutions of binomial edge ideals; study the asymptotic structure of minimal free resolutions over exterior algebras.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.